In Part 2 of the base study (Week 12 to Week 28), participants in Arm A, Arm B, and Arm D will receive matching placebo to MK-3222 to maintain blinding at Week 12. Participants in Arm A and Arm B will continue to receive either MK-3222 200 mg (Arm A) or MK-3222 100 mg (Arm B) at Week 16 and Week 28 and will also receive matching placebo to etanercept once weekly through study Week 28. At study Week 12, Arm C participants will be re-randomized to receive their first dose of MK-3222 200 mg or MK-3222 100 mg, and will receive additional doses of study medication according to their re-randomized treatment assignment at Week 16 and Week 28. Participants in Arm C will also receive matching placebo to etanercept once weekly through treatment Week 28. Participants in Arm D will continue with once weekly doses of etanercept through study Week 28 in combination with matching placebo to MK-3222.